Problem: Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the solutions to
\begin{align*}
|x - 3| &= |y - 9|, \\
|x - 9| &= 2|y - 3|.
\end{align*}Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$
Answer: From the given equations,
\begin{align*}
(x - 3) &= \pm (y - 9), \\
(x - 9) &= \pm 2 (y - 3).
\end{align*}Thus, we divide into cases.

Case 1: $x - 3 = y - 9$ and $x - 9 = 2(y - 3).$

Solving this system, we find $(x,y) = (-15,-9).$

Case 2: $x - 3 = y - 9$ and $x - 9 = -2(y - 3).$

Solving this system, we find $(x,y) = (1,7).$

Case 3: $x - 3 = -(y - 9)$ and $x - 9 = 2(y - 3).$

Solving this system, we find $(x,y) = (9,3).$

Case 4: $x - 3 = -(y - 9)$ and $x - 9 = -2(y - 3).$

Solving this system, we find $(x,y) = (9,3).$

Hence, the solutions $(x,y)$ are $(-15,-9),$ $(1,7),$ and $(9,3).$  The final answer is $(-15) + (-9) + 1 + 7 + 9 + 3 = \boxed{-4}.$